Learning to Bound: A Generative Cram\'er-Rao Bound

TL;DR

This paper introduces a data-driven method to approximate the Cramér-Rao bound using deep generative models, enabling performance bounds estimation without explicit statistical models, demonstrated on image processing tasks.

Contribution

It proposes the Generative Cramér-Rao Bound (GCRB), a novel approach that leverages deep generative models to estimate bounds without requiring analytical likelihoods.

Findings

GCRB accurately approximates the CRB in simple problems.

GCRB effectively estimates bounds in image denoising tasks.

The method benefits from deep generative models' ability to capture complex data distributions.

Abstract

The Cram\'er-Rao bound (CRB), a well-known lower bound on the performance of any unbiased parameter estimator, has been used to study a wide variety of problems. However, to obtain the CRB, requires an analytical expression for the likelihood of the measurements given the parameters, or equivalently a precise and explicit statistical model for the data. In many applications, such a model is not available. Instead, this work introduces a novel approach to approximate the CRB using data-driven methods, which removes the requirement for an analytical statistical model. This approach is based on the recent success of deep generative models in modeling complex, high-dimensional distributions. Using a learned normalizing flow model, we model the distribution of the measurements and obtain an approximation of the CRB, which we call Generative Cram\'er-Rao Bound (GCRB). Numerical experiments on simple problems validate this approach, and experiments on two image processing tasks of image denoising and edge detection with a learned camera noise model demonstrate its power and benefits.